I am trying to solve the following exercise from Set Theory(Kunen, 2011):
Assume $\mathit{MA}(\kappa)$. Fix $\tau < \omega_{1}$ and $\mathcal{F} \subseteq [\omega_{1}]^{\aleph_{0}}$ such that $|\mathcal{F}| \leq \kappa$ and such that all sets in $\mathcal{F}$ have order type $< \tau$. Prove that there is an uncountable $J \subseteq \omega_{1}$ such that $J \cap x$ is finite for all $x \in \mathcal{F}$.
The natural poset to consider is:
\begin{equation*}
\mathbb{P} = \{p : p = \langle A_{p},B_{p}\rangle \text{ where } A_{p} \in [\omega_{1}]^{<\omega} \wedge B_{p} \in [\mathcal{F}]^{<\omega}\}
\end{equation*}
Declare $p \leq q \iff A_{p} \supseteq A_{q} \wedge B_{p} \supseteq B_{q} \wedge \forall x \in B_{q}[x \cap A_{p}{\setminus}A_{q}= \emptyset]$.
Assuming that $\mathbb{P}$ has the ccc, it is easy to see that such a $J$ exists using $\mathit{MA}(\kappa)$. However, I can't prove that $\mathbb{P}$ has the ccc.
Here is my attempt: Suppose to the contrary that there is an uncountable antichain $\{p_{\alpha}: \alpha < \omega_{1}\}$. We can assume WLOG that there is an $n \in \omega$ such that for each $\alpha < \omega_{1}$, $|A_{p_{\alpha}}| = n$. By the Delta-System Lemma and cutting down if necessary, we can assume WLOG that there is an $R \in [\omega_{1}]^{<\omega}$ and an $S \in [\mathcal{F}]^{<\omega}$ such that for each $\alpha,\beta < \omega_{1}$ with $\alpha \not = \beta$ we have $A_{p_{\alpha}} \cap A_{p_{\beta}} = R$ and $B_{p_{\alpha}} \cap B_{p_{\beta}} = S$. We can also assume WLOG that $R = \emptyset$. Also, since $\bigcup S$ is countable we can have $A_{p_{\alpha}} \cap \bigcup S = \emptyset$ for large enough $\alpha$. So replacing $B_{p_{\alpha}}$ by $B_{p_{\alpha}}{\setminus}S$, we can assume WLOG that $S = 0$. Again for large enough $\alpha$ we have $A_{p_{\alpha}} \cap \bigcup_{\beta < \tau} \bigcup B_{p_{\beta}} = \emptyset$ and so WLOG assume that for all $\tau \leq \alpha < \omega_{1}$, we have $A_{p_{\alpha}} \cap \bigcup_{\beta < \tau}\bigcup B_{p_{\beta}} = \emptyset$. I feel that there should there should be a $\beta < \tau$ such that $A_{p_{\beta}} \cap \bigcup B_{p_{\tau}} = \emptyset$. Is this even true? Are there any other approaches?
Best Answer
Using the $\Delta$-system Lemma to make the conditions a bit more uniform is always a good start. Let's assume you did all that, as you described above.
For $\alpha < \omega_1$, there might be uncountably many $\beta > \alpha$ so that the reason that $p_\alpha$ and $p_\beta$ are incompatible is that $F \cap A_\beta \setminus A_\alpha \neq \emptyset$ for some $F \in B_\alpha$. Can you finish the argument from here?
Maybe for no $\alpha$, there are uncountably many such $\beta$. Can you construct an uncountable set $X \subseteq \omega_1$ such that for any $\alpha < \beta \in X$, the reason that $p_\alpha$ and $p_\beta$ are incompatible is that $F \cap A_\alpha \setminus A_\beta \neq \emptyset$ for some $F \in B_\beta$? Let $\beta \in X$ be such that $X \cap \beta$ has an indecomposable ordertype strictly larger than $\tau$. For every $\alpha \in X \cap \beta$ there is $F \in B_\beta$ such that $A_\alpha \setminus A_\beta \cap F \neq \emptyset$. Can you finish from here?